Then any irreducible spherical unitary representation of \(G<G\times G\)  with replacing \(\hat{\Delta }\)  with \(\hat{\Delta }^{\mathrm {cop}}\)  and any spherical vector in it must be of the form \(\pi _U\otimes \pi _{\bar{U}} : W^*(G\times G) \curvearrowright \mathcal {H}_U\otimes \mathcal {H}_{\bar{U}}\)  and a scalar multiple of \(\bar{R}_U(1) \in \mathcal {H}_U\otimes \mathcal {H}_{\bar{U}}\) , respectively. (See [1]} and its preliminary discussion.) Consequently, the \(\bar{R}\) -part of solutions of conjugate equations can be obtained as \((\vartheta ^t,+1)\) -spherical representations. It may be possible to give further deformations of diagonal embedding \(\hat{\Delta } : \mathcal {U}(G) \rightarrow \mathcal {U}(G\times G)\) .
